WIP: Hacky(!) but faster nbody implementations

Author: smallnamespace

nbody/nbody_unsafe_simd.jl

@@ -0,0 +1,226 @@
+# Based on https://benchmarksgame-team.pages.debian.net/benchmarksgame/program/nbody-rust-7.html
+#
+# The basic strategy matches Rust #7, based on gcc #4: use vectorized rsqrt
+# to compute pairwise distances.
+#
+# We deviate by also skipping a single Newton step
+
+module NBody
+
+using StaticArrays, SIMD, Printf
+using Base: llvmcall
+
+const solar_mass = 4π^2
+const days_per_year = 365.24
+const NBODIES = 5
+const NPAIRS = Int(NBODIES * (NBODIES - 1) / 2)
+const PAIRS = Tuple((i,j) for i = 1:5, j = 1:5 if j > i)
+
+struct Bodies
+    x::MMatrix{NBODIES, 3, Float64}
+    v::MMatrix{NBODIES, 3, Float64}
+    m::NTuple{NBODIES, Float64}
+end
+
+function init_bodies!(bodies)
+  x, v = bodies.x, bodies.v
+  # Sun
+  x[1, :] = [0, 0, 0]
+  v[1, :] = [0, 0, 0]
+
+  # Jupiter
+  x[2, :] = [
+    4.84143144246472090e+00,
+    -1.16032004402742839e+00,
+    -1.03622044471123109e-01,
+  ]
+  v[2, :] = [
+    1.66007664274403694e-03,
+    7.69901118419740425e-03,
+    -6.90460016972063023e-05,
+  ] .* days_per_year
+
+  # Saturn
+  x[3, :] = [
+    8.34336671824457987e+00,
+    4.12479856412430479e+00,
+    -4.03523417114321381e-01,
+    ]
+  v[3, :] = [
+    -2.76742510726862411e-03,
+    4.99852801234917238e-03,
+    2.30417297573763929e-05,
+  ] .* days_per_year
+
+  # Uranus
+  x[4, :] = [
+    1.28943695621391310e+01,
+    -1.51111514016986312e+01,
+    -2.23307578892655734e-01,
+  ]
+  v[4, :] = [
+    2.96460137564761618e-03,
+    2.37847173959480950e-03,
+    -2.96589568540237556e-05,
+  ] .* days_per_year
+
+  # Neptune
+  x[5, :] = [
+    1.53796971148509165e+01,
+    -2.59193146099879641e+01,
+    1.79258772950371181e-01,
+  ]
+  v[5, :] = [
+    2.68067772490389322e-03,
+    1.62824170038242295e-03,
+    -9.51592254519715870e-05,
+  ] * days_per_year
+end
+
+const __m128 = NTuple{4, VecElement{Float32}}
+const __m128d = NTuple{2, VecElement{Float64}}
+const v2d = Vec{2, Float64}
+
+@inline function rsqrt_pd(v2::v2d)
+    v2d(rsqrt_ccall(v2.elts))
+end
+
+@inline function rsqrt_pd_newton(v2::v2d)
+    guess = rsqrt_pd(v2)
+    # We only need one Newton step to achieve desired accuracy
+    guess = guess * 1.5 - ((0.5 * v2) * guess) * (guess * guess)
+    guess
+end
+
+rsqrt(f::__m128) = ccall("llvm.x86.sse.rsqrt.ps", llvmcall, __m128, (__m128, ), f);
+_mm_cvtpd_ps(f::__m128d) = ccall("llvm.x86.sse2.cvtpd2ps", llvmcall, __m128, (__m128d, ), f);
+_mm_cvtps_pd(f::__m128) = llvmcall(("", "
+              %2 = shufflevector <4 x float> %0, <4 x float> undef, <2 x i32> <i32 0, i32 1>
+              %3 = fpext <2 x float> %2 to <2 x double>
+              ret <2 x double> %3"),
+        __m128d,
+        Tuple{__m128}, f)
+@inline rsqrt_ccall(f::__m128d) = _mm_cvtps_pd(rsqrt(_mm_cvtpd_ps(f)))
+
+@inline function advance(#x, v, m, dt, dx, dmag)
+    x::MMatrix{NBODIES, 3, Float64, NBODIES * 3},
+    v::MMatrix{NBODIES, 3, Float64, NBODIES * 3},
+    m::NTuple{NBODIES, Float64},
+    dt::Float64,
+    dx::MMatrix{NPAIRS, 3, Float64, NPAIRS * 3},
+    dmag::MVector{NPAIRS, Float64})
+
+    dmag_v2d_ptr = Base.unsafe_convert(Ptr{v2d}, pointer_from_objref(dmag))
+    dx_v2d_ptr = Base.unsafe_convert(Ptr{v2d}, pointer_from_objref(dx))
+
+    # Unroll loop to calculate distances + store two at a time
+    @inbounds for k1 = 1:2:length(PAIRS)
+        k2 = k1 + 1
+        k_v2d = k2 ÷ 2
+
+        i1, j1 = PAIRS[k1]
+        i2, j2 = PAIRS[k2]
+
+        dx1 = v2d((x[i1, 1], x[i2, 1])) - v2d((x[j1, 1], x[j2, 1]))
+        dx2 = v2d((x[i1, 2], x[i2, 2])) - v2d((x[j1, 2], x[j2, 2]))
+        dx3 = v2d((x[i1, 3], x[i2, 3])) - v2d((x[j1, 3], x[j2, 3]))
+        unsafe_store!(dx_v2d_ptr, dx1, k_v2d)
+        unsafe_store!(dx_v2d_ptr, dx2, k_v2d + NPAIRS ÷ 2)
+        unsafe_store!(dx_v2d_ptr, dx3, k_v2d + NPAIRS)
+
+        dsq = dx1^2 + dx2^2 + dx3^2
+        drsqrt = rsqrt_pd_newton(dsq)
+        mag = dt * drsqrt / dsq
+        unsafe_store!(dmag_v2d_ptr, mag, k_v2d)
+    end
+
+    k = 1
+    @inbounds for (i, j) = PAIRS
+        dmag_i = dmag[k] * m[i]
+        dmag_j = dmag[k] * m[j]
+        for d = 1:3
+          dx_k = dx[k, d]
+          v[i, d] -= dx_k * dmag_j
+          v[j, d] += dx_k * dmag_i
+        end
+        k += 1
+    end
+
+    @inbounds for i = 1:NBODIES
+      for d = 1:3
+        x[i, d] += dt * v[i, d]
+      end
+    end
+end
+
+function energy(bodies)
+  x, v, m = bodies.x, bodies.v, bodies.m
+  e = 0.0
+  for i = 1:NBODIES
+    e += 0.5 * m[i] * sum(v[i, :].^2)
+    for j = i + 1:NBODIES
+      dx = x[i, :] - x[j, :]
+      distance = sqrt(sum(dx .* dx))
+      e -= (m[i] * m[j]) / distance
+    end
+  end
+  return e
+end
+
+function init_sun!(bodies)
+  px = [0.0, 0.0, 0.0]
+  for i = 1:NBODIES
+    px += bodies.v[i, :] * bodies.m[i]
+  end
+  bodies.v[1, :] = -px ./ solar_mass
+end
+
+function main(iterations::Int64)
+  n = iterations
+
+  x = zeros(MMatrix{NBODIES, 3, Float64, 15})
+  v = zeros(MMatrix{NBODIES, 3, Float64, 15})
+  m = NTuple{NBODIES, Float64}((
+    1.0,
+    9.54791938424326609e-04,
+    2.85885980666130812e-04,
+    4.36624404335156298e-05,
+    5.15138902046611451e-05,
+  ) .* solar_mass)
+  bodies = Bodies(x, v, m)
+
+  init_bodies!(bodies)
+  init_sun!(bodies)
+  @printf("%.9f\n", energy(bodies))
+
+  # Buffers
+  dx = zeros(MMatrix{NPAIRS, 3, Float64, 30})
+  dmag = zeros(MVector{NPAIRS, Float64})
+  for _ = 1:n
+    advance(x, v, m, 0.01, dx, dmag)
+  end
+
+  @printf("%.9f\n", energy(bodies))
+end
+
+end
+
+@time NBody.main(parse(Int64, ARGS[1]))
+@time NBody.main(parse(Int64, ARGS[1]))
+
+# > julia -O3 -C core2 -- nbody_unsafe_simd.jl 50000000
+# -0.169075164
+# -0.169060076
+#   7.603160 seconds (7.88 M allocations: 397.438 MiB, 1.82% gc time)
+# -0.169075164
+# -0.169060076
+#   4.172609 seconds (470 allocations: 11.891 KiB)
+
+# using StaticArrays, InteractiveUtils
+# code_native(nb.advance,
+#   (MMatrix{nb.NBODIES, 3, Float64, nb.NBODIES * 3},
+#   MMatrix{nb.NBODIES, 3, Float64, nb.NBODIES * 3},
+#   NTuple{nb.NBODIES, Float64},
+#   Float64,
+#   MMatrix{nb.NPAIRS, 3, Float64, nb.NPAIRS * 3},
+#   MVector{nb.NPAIRS, Float64}))